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Given the set od propagators (say momenta flows) on a Feynman diagram (flow network), I would like to decide whether this diagram is planar or not.

I know that non-planar diagrams manifest different "mixing" between independent propagators (for different loops), but I can not find any general rule, hence my question: are there any flow criteria for a graph being planar or not?

Edit: I found out that network flows are also called "weighted graphs", and the idea crossed my mind: if there could be defined a notion of "independent weights" of loops (considering graph theory, these are called circuits or cycles perhaps), the mixing of such weights would probably give the demanded distinction. Maybe some resolution of a graph to independent cycles (not in common sense, but rather with respect to their weight) would be necessary?

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    $\begingroup$ By Kuratowski's theorem, I can hardly imagine that the flow network structure may play any role in the planarity of the underlying graph - unless one introduces any form of oriented and weighted planarity, which I am unaware of. $\endgroup$ Jan 29, 2013 at 16:29

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